If three circles “kiss,” like the black circles above, then a fourth circle can be drawn that’s tangent to all three. In 1643 René Descartes showed that if the curvature or “bend” of a circle is defined as k = 1/r, then the radius of the fourth circle can be found by

The ± sign reflects the fact that two solutions are generally possible — the plus sign corresponds to the smaller red circle, the minus sign to the larger (circumscribing) one.

Frederick Soddy summed this up in a poem in Nature (June 20, 1936):

The Kiss Precise

For pairs of lips to kiss maybe
Involves no trigonometry.
‘Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

Horses are difficult to manage onstage, so in 1901 music hall performer Alexander Braatz invented this ingenious alternative. The actor’s own legs masquerade as those of his horse, and a system of levers and cords moves the hind legs as well.

False human legs are attached at each side to give the appearance that the performer is mounted; his real feet “are advantageously covered by large clumsy hoof-like coverings.” I suppose you could even race these things between rehearsals.

A gentleman, who had been described as a ‘Pettifogger,’ accused another gentleman, whom he had styled a ‘Fish-fag,’ with an assault. It being a very intricate point, it was of course referred to the Lord Mayor. It stood as follows: — ‘Whether puffing a cloud of tobacco-smoke in a man’s face constituted an assault?’ After some grave consultation with that encyclopaedia of wisdom, Mr. Hobler, the decision ran thus — The Lord Mayor: ‘There has been no assault; nothing but words, words.’ — Complainant: ‘I beg pardon, my Lord.’ — The Lord Mayor: ‘Well, then, all smoke, if you please, or words and puffs. There have been no blows.’ — Now we beg his Lordship’s pardon. Pray what is a puff but a blow?

Consider them in turn. A can’t be true because some of the other statements contradict one another (for example, C and D). B can’t be true because it implies that C is also true. Since A and B are false, C is also false, and likewise D. But E is true, and this makes F false. So the answer is E.

Stewardess Violet Jessop was both cursed and blessed — during the 1910s she met disaster on all three of the White Star Line’s Olympic class of gigantic ocean liners, but she managed to escape each time.

In this episode of the Futility Closet podcast we’ll accompany Violet on her three ill-fated voyages, including the famous sinkings of the Titanic and the Britannic, and learn the importance of toothbrushes in ocean disasters.

We’ll also play with the International Date Line and puzzle over the identity of Salvador Dalí’s brother.

University of Chicago economist Steven Levitt discusses his coin-flipping experiment about halfway through this BBC podcast. The associated website is here.

We first wrote about Violet Jessop on March 11, 2009. Maritime historian John Maxtone-Graham interviewed her in 1970 for The Only Way to Cross, his 1978 book about the era of ocean liners. When Violet died in 1971 she left a manuscript to her daughters, which, edited by Maxtone-Graham, came to light in 1997 as Titanic Survivor: The Newly Discovered Memoirs of Violet Jessop, Who Survived Both the Titanic and Britannic Disasters. A poetic note from Maxtone-Graham in that book:

“One particular service commemorates the 1500 lost on the Titanic: Every 14th of April, a United States Coast Guard cutter comes to pay the homage of the Ice Patrol, which owes its inception to the disaster. With engines stilled and church pennant at the masthead, officers and men line the deck in full dress, while the commander reads the burial service. Three volleys of rifle fire can be heard, then the cutter passes on, leaving a lone wreath on the waves above the broken hull.”

Lewis Carroll underscored the need for an international date line with this conundrum, which he presented among the mathematical puzzle stories he wrote for the Monthly Packet in the 1880s:

The day changes only at midnight. Suppose it’s midnight in Chelsea; Wednesday has concluded and Thursday is about to begin. It’s still Wednesday in Ireland and America, and it’s already Thursday in Germany and Russia.

That’s fine. But continue in both directions. If it’s Wednesday in America, is it Wednesday in Hawaii? If it’s Thursday in Russia, is it Thursday in Japan? Mustn’t the two days “meet” on the farther side of the globe?

“It isn’t midnight anywhere else; so it can’t be changing from one day to another anywhere else. And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there must be some place, not Chelsea, that has different days on the two sides of it. And the worst of it is, the people there get their days in the wrong order: they’ve got Wednesday east of them, and Thursday west — just as if their day had changed from Thursday to Wednesday!”

Carroll normally presented the solution to each problem in the following month’s number. In this case he postponed the solution, “partly because I am myself so entirely puzzled by it,” and then discontinued the column without resolving the problem.

Caroline leaves town driving at a constant speed. After some time she passes a milestone displaying a two-digit number. An hour later she passes a milestone displaying the same two digits but in reverse order. In another hour she passes a third milestone with the same two digits (in some order) separated by a zero.

The number on the first milestone can be expressed as 10A + B, and the number on the second milestone is 10B + A. The number on the third milestone is either 100A + B or 100B + A.

Caroline’s speed is constant, so the distances between the milestones are equal. What’s the greatest this distance could be? Well, setting the two digits as far apart as possible would mean that one digit is 1 and the other is 9; then the first milestone would be at 19, the second at 91, and the third at 163. This doesn’t satisfy the puzzle, but it tells us that the number in the hundreds place must be 1 — Caroline doesn’t get as far as the 200-mile mark.

That means that either 100A or 100B must equal 100. Which is it? It must be 100A, since the number on the first milestone, where A stands for tens, is smaller than the number on the second milestone, where B stands for tens. So A is 1, the first milestone is 10 + B, the second milestone is 10B + 1, and the third milestone is 100 + B. Since the distances between milestones are equal, we have:

(10B + 1) – (10 + B) = (100 + B) – (10B + 1)
18B = 108
B = 6

A = 1 and B = 6, so the milestones carry the numbers 16, 61, and 106. Caroline’s speed is 45 mph.

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Futility Closet is a collection of entertaining curiosities in history, literature, language, art, philosophy, and mathematics, designed to help you waste time as enjoyably as possible.

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